Abstract: This paper explores the existence of infinite order meromorphic functions and algebroidal functions with given T directions. An algebroidal function $ \omega(z) $ of infinite order was constructed such that for an arbitrary given non-empty closed set of real numbers $ E\pmod {2\pi} $, $\{z: \arg z = \theta, \theta\in E\}$ is the set of all T directions and Borel directions of infinite order of $ \omega(z) $; using a method of cyclic values-assigned, the distribution of singular directions for meromorphic functions and algebroidal functions of infinite order is obtained.

Key words: algebroidal function, infinite order, T direction, Borel direction